# HW2.3 ## Advection-Diffusion and N-S equation --- - Advection-Diffusion equation - N-S equation - Compare --- ## Advection-Diffusion equation --- ## The equation $\quad\quad \dfrac{\partial u}{\partial t} + u\,\dfrac{\partial u}{\partial x} = D\dfrac{\partial^2 u}{\partial x^2}$ - u: Thing we interesting - D: The diffusion coefficient --- ## Derivation - The general equation $\quad\quad \dfrac{\partial u}{\partial t} =\mathbf{\nabla} \cdot (D \mathbf{\nabla} u) - \mathbf{\nabla} \cdot (\mathbf{v} u) + R$ $\mathbf{\nabla} \cdot (D \mathbf{\nabla} u)$:The diffusion term. $\quad$$\mathbf{\nabla} \cdot (\mathbf{v} u)$:The convection term. $\quad\quad\quad$ $R$:lose $\quad\quad\quad$ v:The volcity flied --- ## assume - D = constant => $\mathbf{\nabla} \cdot (D \mathbf{\nabla} u)=D\nabla^2u$ - incompressible flow =>$\mathbf{\nabla} \cdot (\mathbf{v} u) = \mathbf{v}\cdot\nabla u$ + $u\nabla \cdot\mathbf{v}$ =>$\mathbf{\nabla} \cdot (\mathbf{v} u) = \mathbf{v}\cdot\nabla u$ - ignore $\mathbf{R}$ --- ## We can get - total $\quad\quad \dfrac{\partial u}{\partial t} + \mathbf{v}\cdot\nabla u = D\nabla^2u$ - Only 1D $\quad\quad \dfrac{\partial u}{\partial t} + \mathbf{u}\,\dfrac{\partial u}{\partial x} = D\dfrac{\partial^2 u}{\partial x^2}$ --- ## N-S equation --- ## The equation $\rho(\dfrac{\partial u}{\partial t}+u\,\dfrac{\partial u}{\partial x}+v\,\dfrac{\partial u}{\partial y}+w\,\dfrac{\partial u}{\partial z})=\\\rho g_x-\dfrac{\partial p}{\partial x}+\mu(\dfrac{\partial^2 u}{\partial x^2}+\dfrac{\partial^2 u}{\partial y^2}+\dfrac{\partial^2 u}{\partial z^2})$ --- ## Each term ##### $\quad\quad\quad\quad\quad\quad\quad\dfrac{\partial u}{\partial t}$:Unsteady acceleration ##### $u\,\dfrac{\partial u}{\partial x}+v\,\dfrac{\partial u}{\partial y}+w\,\dfrac{\partial u}{\partial z}$:Convective acceleration ##### $\quad\quad\quad\quad\quad\quad\quad\quad g_x$:gravity ##### $\quad\quad\quad\quad\quad\quad\quad-\dfrac{\partial p}{\partial x}$:Pressure gradient ##### $\mu(\dfrac{\partial^2 u}{\partial x^2}+\dfrac{\partial^2 u}{\partial y^2}+\dfrac{\partial^2 u}{\partial z^2})$:Viscosity --- ## compare --- - Describe different things - variable - meth tern --- ## only 1D $\quad \dfrac{\partial u}{\partial t} + u\,\dfrac{\partial u}{\partial x} = D\dfrac{\partial^2 u}{\partial x^2}$ $(\dfrac{\partial u}{\partial t}+{u}\,\dfrac{\partial u}{\partial x})=\dfrac{\mu}{\rho}(\dfrac{\partial^2 u}{\partial x^2}+\rho g_x-\dfrac{\partial p}{\partial x})$ --- $\quad\quad\quad\quad\quad\quad$N-s:$\mathbf{u}\cdot\nabla\mathbf{u}$ Advection-Diffusion:$\mathbf{v}\cdot\nabla u$